Furthermore, data on highly dense stellar objects share with the oscillating nEDM signal its model-independence character: they directly provide constraints on the \(\lbrace m_a, f_a\rbrace \)  parameter space. Those data have the added value of not relying on an axionic nature for DM.
They have already set strong constraints on the \(\lbrace m_a, f_a\rbrace \)  parameter space [1]}, [2]}, [3]}. For the \(Z_{\cal N}\)  axion scenario under discussion, they implied a strict limit on the number of mirror worlds: \(3\le {\cal N}\lesssim 47\)  for \(f_a\lesssim 2.4\times 10^{15}\)  GeV. They also provide tantalizing
discovery prospects for any value of \(f_a\)  and down to \({\cal N}\sim 9\)  –possibly even lower–
with future neutron star and gravitational wave data, down to the ultra-light mass region, see Fig. 8 in Ref. [3]}. It is thus of crucial interest to compare them with the oscillating nEDM projects described above. Fig. REF  tends to this task and shows that present finite-density data leave wide open the complete parameter space that CASPEr-Electric I, II and III can cover, with exciting synergy prospects between both type of approaches.
